 reserve U for set,
         X, Y for Subset of U;
 reserve U for non empty set,
         A, B, C for non empty IntervalSet of U;

theorem
  A _/\_ (B _\/_ C) = (A _/\_ B) _\/_ (A _/\_ C)
  proof
A1:  A _/\_ (B _\/_ C) c= (A _/\_ B) _\/_ (A _/\_ C)
     proof
       let x be object;
       assume x in A _/\_ (B _\/_ C); then
       consider X, Y being set such that
A2:    X in A & Y in UNION (B,C) & x = X /\ Y by SETFAM_1:def 5;
       consider Z, W being set such that
A3:    Z in B & W in C & Y = Z \/ W by A2,SETFAM_1:def 4;
A4:    A is non empty ordered Subset-Family of U &
         B is non empty ordered Subset-Family of U &
         C is non empty ordered Subset-Family of U by Lm4;
       X /\ (Z \/ W) in INTERSECTION (A, UNION (B,C)) by A2,A3,SETFAM_1:def 5;
       hence thesis by A2,A3,Th30,A4;
     end;
    (A _/\_ B) _\/_ (A _/\_ C) c= A _/\_ (B _\/_ C)
    proof
      let x be object;
      assume x in (A _/\_ B) _\/_ (A _/\_ C); then
      consider X,Y being set such that
A5:   X in INTERSECTION (A,B) & Y in INTERSECTION (A,C) & x = X \/ Y
        by SETFAM_1:def 4;
A6:   A is non empty ordered Subset-Family of U & B is non empty ordered
      Subset-Family of U & C is non empty ordered Subset-Family of U by Lm4;
      x in UNION (INTERSECTION (A,B), INTERSECTION (A,C))
        by A5,SETFAM_1:def 4;
      hence thesis by Th30,A6;
    end;
    hence thesis by A1;
 end;
